Stability Analysis and Stabilization of Randomly Switched Systems
Stability Analysis and Stabilization of Randomly Switched Systems D. Chatterjee and D. Liberzon dchatter, [email protected]
Coordinated Science Laboratory University of Illinois at Urbana-Champaign Thursday, December 14, 2006 CDC’06 ThA09: Stochastic and Discrete-Time Switched Systems
Abstract We present sufficient conditions for almost sure global asymptotic stability (gas a.s.) of randomly switched systems via multiple Lyapunov-like functions. For systems possessing control inputs we present a method for designing controllers which render the closed-loop randomly switched system gas a.s.
1.
The Analysis Problem
System: x˙ = fσ (x),
(x(0), σ(0)) = (x0 , σ0 ),
t>0
• x ∈ Rn , fp vector field on Rn , fp (0) = 0, p ∈ P—finite index set • switching signal σ is a P-valued random process
(?)
1.
The Analysis Problem
System: x˙ = fσ (x),
(x(0), σ(0)) = (x0 , σ0 ),
t>0
• x ∈ Rn , fp vector field on Rn , fp (0) = 0, p ∈ P—finite index set • switching signal σ is a P-valued random process Global asymptotic stability almost surely (gas a.s.): System (?) is gas a.s. if • ∀ ε > 0 ∃ δ > 0 : |x0 | < δ ⇒ P |x(t)| < ε ∀ t > 0 = 1 • ∀ r, ε0 > 0 ∃ T > 0 : |x0 | < r ⇒ P |x(t)| < ε0 ∀ t > T = 1 Aim: Find sufficient conditions for gas a.s. of (?)
(?)
1.
The Analysis Problem
System: x˙ = fσ (x),
(x(0), σ(0)) = (x0 , σ0 ),
t>0
• x ∈ Rn , fp vector field on Rn , fp (0) = 0, p ∈ P—finite index set • switching signal σ is a P-valued random process Global asymptotic stability almost surely (gas a.s.): System (?) is gas a.s. if • ∀ ε > 0 ∃ δ > 0 : |x0 | < δ ⇒ P |x(t)| < ε ∀ t > 0 = 1 • ∀ r, ε0 > 0 ∃ T > 0 : |x0 | < r ⇒ P |x(t)| < ε0 ∀ t > T = 1 Aim: Find sufficient conditions for gas a.s. of (?) Approach: • extract properties of individual modes (via Lyapunov-like functions) • extract statistical properties of switching signal σ • connect the two sets of properties
(?)
2.
Analysis Results: all modes stable
System:
(?)
x˙ = fσ (x)
Theorem A.[To appear: IEEE TAC] System (?) is gas a.s. if (G1) ∃ C 1 pos def rad unbdd Vp : Rn → R>0 , ∃ µ > 1, λ◦ > 0: (V1) Lfp Vp (x) 6 −λ◦ Vp (x)
(V2) Vp1 (x) 6 µVp2 (x)
e>0: (G2) ∃ λ, λ
Nσ (t) = # switches on [0, t[
k e P Nσ (t) = k 6 e−λt λt /k! ∀ t, ∀ k
e + λ◦ /λ (G3) µ < λ
2.
Analysis Results: all modes stable
System:
(?)
x˙ = fσ (x)
Theorem A.[To appear: IEEE TAC] System (?) is gas a.s. if (G1) ∃ C 1 pos def rad unbdd Vp : Rn → R>0 , ∃ µ > 1, λ◦ > 0: (V1) Lfp Vp (x) 6 −λ◦ Vp (x)
(V2) Vp1 (x) 6 µVp2 (x)
e>0: (G2) ∃ λ, λ
Nσ (t) = # switches on [0, t[
k e P Nσ (t) = k 6 e−λt λt /k! ∀ t, ∀ k
e + λ◦ /λ (G3) µ < λ Remarks: • λ◦ > 0 (no loss of generality) ⇔ every mode stable • µ standard in deterministic results, but restrictive • (G2) is loose description—no transition probabilities involved • (G2) ⇒ statistically slow switching • Continuous-time (π ◦ , Q)–Markov chains satisfy (G2) with λ := max|qii |, i∈P
e := max qij λ i,j∈P
3.
Analysis Results: some unstable modes
System:
x˙ = fσ (x)
(?)
3.
Analysis Results: some unstable modes
System:
x˙ = fσ (x)
Remarks: • Just slow switching insufficient • Need to keep track of active periods of bad modes
(?)
3.
Analysis Results: some unstable modes
System:
x˙ = fσ (x)
Remarks: • Just slow switching insufficient • Need to keep track of active periods of bad modes Theorem B. System (?) is gas a.s. if (H1) ∃ C 1 pos def rad unbdd Vp : Rn → R>0 , ∃ µ > 1, λp ∈ R: (V1) Lfp Vp (x) 6 −λp Vp (x) (V2) Vp1 (x) 6 µVp2 (x) (H2) σ satisfies (for example): (S1) (Si )i∈N , Si := τi − τi−1 , is an i.i.d exponential(λ)1 sequence (S2) (σ(τi ))i∈N is an i.i.d sequence, with P σ(τ1 ) = p = qp , p ∈ P (S3) (Si )i∈N is independent of (σ(τi ))i∈N P (H3) λ + λp > 0 and p∈P µqp 1 + λp /λ < 1
1
Density function fSi (s) = λe−λs if s > 0, and 0 else
(?)
3.
Analysis Results: some unstable modes
System:
x˙ = fσ (x)
Remarks: • Just slow switching insufficient • Need to keep track of active periods of bad modes Theorem B. System (?) is gas a.s. if (H1) ∃ C 1 pos def rad unbdd Vp : Rn → R>0 , ∃ µ > 1, λp ∈ R: (V1) Lfp Vp (x) 6 −λp Vp (x) (V2) Vp1 (x) 6 µVp2 (x) (H2) σ satisfies (for example): (S1) (Si )i∈N , Si := τi − τi−1 , is an i.i.d exponential(λ)1 sequence (S2) (σ(τi ))i∈N is an i.i.d sequence, with P σ(τ1 ) = p = qp , p ∈ P (S3) (Si )i∈N is independent of (σ(τi ))i∈N P (H3) λ + λp > 0 and p∈P µqp 1 + λp /λ < 1 Note: • λ fixed ⇒ λp > −λ ∀ p (maximal allowable instability) • λp → −λ (greater instability) ⇒ qp → 0 1
Density function fSi (s) = λe−λs if s > 0, and 0 else
(?)
3.
Analysis Results: some unstable modes
(H1): (V1) Lfp Vp (x) 6 −λp Vp (x) (V2) Vp1 (x) 6 µVp2 (x)
(H2):
(H3):
(S1) (Si )i is i.i.d exp(λ)
(S2) (σ(τi ))i is i.i.d , P σ(τ1 ) = p = qp (S3) (Si )i is independent of (σ(τi ))i
A glimpse into the proof of Theorem B:
• λ + λp > 0 X µqp • 0 X µqp • 0 X µqp • 0 X µqp • 0 X µqp • 0
(†)
i=1
• x ∈ Rn , fp , gp,i vector fields on Rn , fp (0) = gp,i (0) = 0, p ∈ P—finite index set • switching signal σ is a P-valued random process • u = [u1 , . . . , um ]T ∈ U ⊂ Rm Control objective: Find u s.t. (†) is gas a.s. in closed-loop
4.
Synthesis Results
System: x˙ = fσ (x) +
m X
gσ,i (x)ui ,
(x(0), σ(0)) = (x0 , σ0 ),
t>0
(†)
i=1
• x ∈ Rn , fp , gp,i vector fields on Rn , fp (0) = gp,i (0) = 0, p ∈ P—finite index set • switching signal σ is a P-valued random process • u = [u1 , . . . , um ]T ∈ U ⊂ Rm Control objective: Find u s.t. (†) is gas a.s. in closed-loop Approach: • Mode-dependent (u = k(σ, x)) − control-Lyapunov-like functions − Artstein-Sontag universal formulae for feedback stabilization − reduce to analysis results
4.
Synthesis Results
System: x˙ = fσ (x) +
m X
gσ,i (x)ui ,
(x(0), σ(0)) = (x0 , σ0 ),
t>0
(†)
i=1
• x ∈ Rn , fp , gp,i vector fields on Rn , fp (0) = gp,i (0) = 0, p ∈ P—finite index set • switching signal σ is a P-valued random process • u = [u1 , . . . , um ]T ∈ U ⊂ Rm Control objective: Find u s.t. (†) is gas a.s. in closed-loop Approach: • Mode-dependent (u = k(σ, x)) − control-Lyapunov-like functions − Artstein-Sontag universal formulae for feedback stabilization − reduce to analysis results • Mode-independent (u = k(x)) − analysis results still usable since they allow unstable modes − search for u: some modes stabilized, others not too destabilized
5.
Work in Progress + Future Directions • Input-to-state disturbance attenuation under random switching • Controller synthesis under partial information of σ • Extending results to Markovian jump systems • Detailed proofs in extended report available at: http://decision.csl.uiuc.edu/~liberzon/publications.html
Coordinated Science Laboratory University of Illinois at Urbana-Champaign Thursday, December 14, 2006 CDC’06 ThA09: Stochastic and Discrete-Time Switched Systems
Abstract We present sufficient conditions for almost sure global asymptotic stability (gas a.s.) of randomly switched systems via multiple Lyapunov-like functions. For systems possessing control inputs we present a method for designing controllers which render the closed-loop randomly switched system gas a.s.
1.
The Analysis Problem
System: x˙ = fσ (x),
(x(0), σ(0)) = (x0 , σ0 ),
t>0
• x ∈ Rn , fp vector field on Rn , fp (0) = 0, p ∈ P—finite index set • switching signal σ is a P-valued random process
(?)
1.
The Analysis Problem
System: x˙ = fσ (x),
(x(0), σ(0)) = (x0 , σ0 ),
t>0
• x ∈ Rn , fp vector field on Rn , fp (0) = 0, p ∈ P—finite index set • switching signal σ is a P-valued random process Global asymptotic stability almost surely (gas a.s.): System (?) is gas a.s. if • ∀ ε > 0 ∃ δ > 0 : |x0 | < δ ⇒ P |x(t)| < ε ∀ t > 0 = 1 • ∀ r, ε0 > 0 ∃ T > 0 : |x0 | < r ⇒ P |x(t)| < ε0 ∀ t > T = 1 Aim: Find sufficient conditions for gas a.s. of (?)
(?)
1.
The Analysis Problem
System: x˙ = fσ (x),
(x(0), σ(0)) = (x0 , σ0 ),
t>0
• x ∈ Rn , fp vector field on Rn , fp (0) = 0, p ∈ P—finite index set • switching signal σ is a P-valued random process Global asymptotic stability almost surely (gas a.s.): System (?) is gas a.s. if • ∀ ε > 0 ∃ δ > 0 : |x0 | < δ ⇒ P |x(t)| < ε ∀ t > 0 = 1 • ∀ r, ε0 > 0 ∃ T > 0 : |x0 | < r ⇒ P |x(t)| < ε0 ∀ t > T = 1 Aim: Find sufficient conditions for gas a.s. of (?) Approach: • extract properties of individual modes (via Lyapunov-like functions) • extract statistical properties of switching signal σ • connect the two sets of properties
(?)
2.
Analysis Results: all modes stable
System:
(?)
x˙ = fσ (x)
Theorem A.[To appear: IEEE TAC] System (?) is gas a.s. if (G1) ∃ C 1 pos def rad unbdd Vp : Rn → R>0 , ∃ µ > 1, λ◦ > 0: (V1) Lfp Vp (x) 6 −λ◦ Vp (x)
(V2) Vp1 (x) 6 µVp2 (x)
e>0: (G2) ∃ λ, λ
Nσ (t) = # switches on [0, t[
k e P Nσ (t) = k 6 e−λt λt /k! ∀ t, ∀ k
e + λ◦ /λ (G3) µ < λ
2.
Analysis Results: all modes stable
System:
(?)
x˙ = fσ (x)
Theorem A.[To appear: IEEE TAC] System (?) is gas a.s. if (G1) ∃ C 1 pos def rad unbdd Vp : Rn → R>0 , ∃ µ > 1, λ◦ > 0: (V1) Lfp Vp (x) 6 −λ◦ Vp (x)
(V2) Vp1 (x) 6 µVp2 (x)
e>0: (G2) ∃ λ, λ
Nσ (t) = # switches on [0, t[
k e P Nσ (t) = k 6 e−λt λt /k! ∀ t, ∀ k
e + λ◦ /λ (G3) µ < λ Remarks: • λ◦ > 0 (no loss of generality) ⇔ every mode stable • µ standard in deterministic results, but restrictive • (G2) is loose description—no transition probabilities involved • (G2) ⇒ statistically slow switching • Continuous-time (π ◦ , Q)–Markov chains satisfy (G2) with λ := max|qii |, i∈P
e := max qij λ i,j∈P
3.
Analysis Results: some unstable modes
System:
x˙ = fσ (x)
(?)
3.
Analysis Results: some unstable modes
System:
x˙ = fσ (x)
Remarks: • Just slow switching insufficient • Need to keep track of active periods of bad modes
(?)
3.
Analysis Results: some unstable modes
System:
x˙ = fσ (x)
Remarks: • Just slow switching insufficient • Need to keep track of active periods of bad modes Theorem B. System (?) is gas a.s. if (H1) ∃ C 1 pos def rad unbdd Vp : Rn → R>0 , ∃ µ > 1, λp ∈ R: (V1) Lfp Vp (x) 6 −λp Vp (x) (V2) Vp1 (x) 6 µVp2 (x) (H2) σ satisfies (for example): (S1) (Si )i∈N , Si := τi − τi−1 , is an i.i.d exponential(λ)1 sequence (S2) (σ(τi ))i∈N is an i.i.d sequence, with P σ(τ1 ) = p = qp , p ∈ P (S3) (Si )i∈N is independent of (σ(τi ))i∈N P (H3) λ + λp > 0 and p∈P µqp 1 + λp /λ < 1
1
Density function fSi (s) = λe−λs if s > 0, and 0 else
(?)
3.
Analysis Results: some unstable modes
System:
x˙ = fσ (x)
Remarks: • Just slow switching insufficient • Need to keep track of active periods of bad modes Theorem B. System (?) is gas a.s. if (H1) ∃ C 1 pos def rad unbdd Vp : Rn → R>0 , ∃ µ > 1, λp ∈ R: (V1) Lfp Vp (x) 6 −λp Vp (x) (V2) Vp1 (x) 6 µVp2 (x) (H2) σ satisfies (for example): (S1) (Si )i∈N , Si := τi − τi−1 , is an i.i.d exponential(λ)1 sequence (S2) (σ(τi ))i∈N is an i.i.d sequence, with P σ(τ1 ) = p = qp , p ∈ P (S3) (Si )i∈N is independent of (σ(τi ))i∈N P (H3) λ + λp > 0 and p∈P µqp 1 + λp /λ < 1 Note: • λ fixed ⇒ λp > −λ ∀ p (maximal allowable instability) • λp → −λ (greater instability) ⇒ qp → 0 1
Density function fSi (s) = λe−λs if s > 0, and 0 else
(?)
3.
Analysis Results: some unstable modes
(H1): (V1) Lfp Vp (x) 6 −λp Vp (x) (V2) Vp1 (x) 6 µVp2 (x)
(H2):
(H3):
(S1) (Si )i is i.i.d exp(λ)
(S2) (σ(τi ))i is i.i.d , P σ(τ1 ) = p = qp (S3) (Si )i is independent of (σ(τi ))i
A glimpse into the proof of Theorem B:
• λ + λp > 0 X µqp • 0 X µqp • 0 X µqp • 0 X µqp • 0 X µqp • 0
(†)
i=1
• x ∈ Rn , fp , gp,i vector fields on Rn , fp (0) = gp,i (0) = 0, p ∈ P—finite index set • switching signal σ is a P-valued random process • u = [u1 , . . . , um ]T ∈ U ⊂ Rm Control objective: Find u s.t. (†) is gas a.s. in closed-loop
4.
Synthesis Results
System: x˙ = fσ (x) +
m X
gσ,i (x)ui ,
(x(0), σ(0)) = (x0 , σ0 ),
t>0
(†)
i=1
• x ∈ Rn , fp , gp,i vector fields on Rn , fp (0) = gp,i (0) = 0, p ∈ P—finite index set • switching signal σ is a P-valued random process • u = [u1 , . . . , um ]T ∈ U ⊂ Rm Control objective: Find u s.t. (†) is gas a.s. in closed-loop Approach: • Mode-dependent (u = k(σ, x)) − control-Lyapunov-like functions − Artstein-Sontag universal formulae for feedback stabilization − reduce to analysis results
4.
Synthesis Results
System: x˙ = fσ (x) +
m X
gσ,i (x)ui ,
(x(0), σ(0)) = (x0 , σ0 ),
t>0
(†)
i=1
• x ∈ Rn , fp , gp,i vector fields on Rn , fp (0) = gp,i (0) = 0, p ∈ P—finite index set • switching signal σ is a P-valued random process • u = [u1 , . . . , um ]T ∈ U ⊂ Rm Control objective: Find u s.t. (†) is gas a.s. in closed-loop Approach: • Mode-dependent (u = k(σ, x)) − control-Lyapunov-like functions − Artstein-Sontag universal formulae for feedback stabilization − reduce to analysis results • Mode-independent (u = k(x)) − analysis results still usable since they allow unstable modes − search for u: some modes stabilized, others not too destabilized
5.
Work in Progress + Future Directions • Input-to-state disturbance attenuation under random switching • Controller synthesis under partial information of σ • Extending results to Markovian jump systems • Detailed proofs in extended report available at: http://decision.csl.uiuc.edu/~liberzon/publications.html
